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Results tagged with ap.analysis-of-pdes
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user 42047
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
10
votes
Differential of a Sobolev map between manifolds
If you are interested, we have given an intrinsic definition of a weak derivative for maps between manifolds A. Convent et J. Van Schaftingen, emphasized Intrinsic colocal weak derivatives and Sobolev …
- 2,824
5
votes
Accepted
Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?
Let $a \in \mathbb{R}^d$ and $r > 0$. We have
$$
\frac{1}{\vert B_r \vert^2}
\int_{B_r} \int_{B_r} \vert I_\alpha (f) (x) - I_\alpha (f) (y) \vert\,\mathrm{d}x\,\mathrm{d}y
\le \frac{c_{d, \alpha} …
- 2,824
4
votes
Accepted
Generalization of maximum principle to other norms
One way to reformulate this is to consider a good extension $\bar{g}$ to the whole $\Omega$ of the function $g$, and then set $v = u - \bar{g}$.
The function $v$ solves then the problem
$$
\left\{
\b …
- 2,824
4
votes
Equivalent Norms on Sobolev Spaces
If $k \in (0, 2]$, we define the multiplier
$$
m (\xi) = (1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k.
$$
We observe that if $\vert \xi \vert \ge 2$, then by differentiability
$$
\big …
- 2,824
4
votes
Example for the Sobolev embedding theorem when p=n.
You can take as an example
$$
u(x) = x_1^{k - 1} (\log \lvert x \rvert)^\beta:
$$
if $\beta < 1 - \frac{1}{n}$, $u \in W^{k,n} (B_1)$ and if $\beta > 0$ then $D^{k - 1} u \not \in L^\infty (B_1)$.
- 2,824
3
votes
"Schwarz symmetrization" on annulus
Such a construction is not possible.
One way to see this is to note that if such a construction was possible, then you would have, by the embeddings for Sobolev spaces of radial functions
$$
\Vert …
- 2,824
3
votes
traces of sobolev spaces under additional assumptions
Partial answer: according to Triebel (Theory of function spaces, 1983, Remark 2.7.5, p. 139), the trace of the Besov space $B^{1/p, p}_1 (\Omega)$ is $L^p (\partial \Omega)$, but the linear extension …
- 2,824
3
votes
functions of bounded variation and gradient vector measure
The property is proved in the litterature (together with its $W^{1, p}$ counterpart):
Augusto C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Different …
- 2,824
3
votes
Accepted
Why $\|u\|_{\tau}\leq C[u]_{W^{s,p}}^a\|u\|_{L^q}^{1-a}$ not correct for $p=1$?
The issue with $p = 1$ is that some definitions of fractional Sobolev spaces that were equivalent when $p > 1$ (by the Gagliardo seminorm that you gave, by interpolation between functional spaces, by …
- 2,824
3
votes
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Here is a direct argument based on the definition by the Gagliardo norm
$$
\Vert u \Vert_{H^{1/2}}^2 = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{\vert u (x) - u (y)\vert^2}{ \vert x - y \vert^{N …
- 2,824
2
votes
Accepted
Hardy-type inequality for point boundary
If $p > n/2$ and if $f \in C^2_c (\mathbb{R}^n \setminus \{0\})$ (twice continuously differentiable functions whose support is compact in $\mathbb{R}^n \setminus \{0\}$), then the weighted Hardy inequ …
- 2,824
2
votes
Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?
By Fubini's theorem and by taking functions that do not depend on the $\mathbb{S}^n$ variable, your estimate is equivalent with the inequality
$$
\int_0^\infty \vert u \vert^2 \le C \int_0^\infty \v …
- 2,824
2
votes
Accepted
Sobolev's lemma on manifolds
This follows from its counterpart in the Euclidean space by local charts.
If you want to have an estimate on the derivative $D^r f$, then you should impose some bounds on derivatives of the curvatur …
- 2,824
2
votes
Sobolev trace map: is the fractional seminorm bounded by just the gradient?
This should follow from the nonhomogeneous trace inequalty
$$
\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2} + \lVert u \rVert_{L^2},
$$
and and from the classical Poincar …
- 2,824
2
votes
Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
I your aim is to apply the Galerkin method, you do not need simultaneous orthonormal basis.
An inspection of Evans’ proof shows that you need a sequence of linear maps $(P_n)_{n \in \mathbb{N}}$ such …
- 2,824